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Section 2.3 Section Summary ! Definition of a Function Section 2.3 Section Summary ! Definition of a Function. ! Domain, Cdomain ! Image, Preimage ! Injection, Surjection, Bijection ! Inverse Function ! Function Composition ! Graphing Functions ! Floor, Ceiling, Factorial ! Partial Functions (optional) Functions Definition: Let A and B be nonempty sets. A function f from A to B, denoted f: A → B is an assignment of each element of A to exactly one element of B. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A. Students Grades ! Functions are sometimes A Carlota Rodriguez called mappings or B . transformations Sandeep Patel C Jalen Williams D F Kathy Scott Functions ! A function f: A → B can also be defined as a subset of A×B (a relation). This subset is restricted to be a relation where no two elements of the relation have the same first element. ! Specifically, a function f from A to B contains one, and only one ordered pair (a, b) for every element a∈ A. and Functions Given a function f: A → B::: ! We say fmapsAto B or f is a mapping from A to B. ! A is called the domain of f. ! B is called the codomain of f. ! If f(a) = b, ! then b is called the image of a under f. ! a is called the preimage of b. ! The range of f is the set of all images of points in A under f. We denote it by f(A). ! Two functions are equal when they have the same domain, the same codomain and map each element of the domain to the same element of the codomain. Representing Functions ! Functions may be specified in different ways: ! An explicit statement of the assignment. Students and grades example. ! A formula. f(x) = x + 1 ! A computer program. ! A Java program that when given an integer n, produces the nth Fibonacci Number (covered in the next section and also inChapter 5). Questions f(a) = ? z AB a The image of d is ? z x b The domain of f is ? A y The codomain of f is ? B c d z The preimage of y is ? b f(A) = ? The preimage(s) of z is (are) ? {a,c,d} Question on Functions and Sets ! If and S is a subset of A, then AB f {a,b,c,} is ? {y,z} a x b f {c,d} is ? {z} y c d z Injections Definition: A function f is said to be one-to-one , or injective, if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. A function is said to be an injection if it is one-to-one. AB a x v b y c z d w Surjections Definition: A function f from A to B is called onto or surjective, if and only if for every element there is an element with . A function f is called a surjection if it is onto. AB a x b y c z d Bijections Definition: A function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto (surjective and injective). AB a x b y c d z w Showing that f is one-to-one or onto Showing that f is one-to-one or onto Example 111: Let f be the function from {a,b,c,d} to {1,2,3} defined by f(a) = 3, f(b) = 2, f(c) = 1, and f(d) = 3. Is f an onto function? Solution: Yes, f is onto since all three elements of the codomain are images of elements in the domain. If the codomain were changed to {1,2,3,4}, f would not be onto. Example 222: Is the function f(x) = x2 from the set of integers onto? Solution: No, f is not onto because there is no integer x with x2 = −1, for example. Inverse Functions Definition: Let f be a bijection from A to B. Then the inverse of f, denoted , is the function from B to A defined as No inverse exists unless f is a bijection. Why? Inverse Functions ABf AB V a a V b b W W c c X d d X Y Y Questions Example 111: Let f be the function from {a,b,c} to {1,2,3} such that f(a) = 2, f(b) = 3, and f(c) = 1. Is f invertible and if so what is its inverse? Solution: The function f is invertible because it is a one-to-one correspondence. The inverse function f-1 reverses the correspondence given by f, so f-1 (1) = c, f-1 (2) = a, and f-1 (3) = b. Questions Example 222:2 Let f: Z ! Z be such that f(x) = x + 1. Is f invertible, and if so, what is its inverse? Solution: The function f is invertible because it is a one-to-one correspondence. The inverse function f-1 reverses the correspondence so f-1 (y) = y – 1. Questions Example 333:3 Let f: R → R be such that . Is f invertible, and if so, what is its inverse? Solution: The function f is not invertible because it is not one-to-one . Composition ! Definition: Let f: B → C, g: A → B. The composition of f with g, denoted is the function from A to C defined by Composition ABg f C AC a a V h h b i b W i c c j d X d j Y Composition Example 111: If and , then and Composition Questions Example 222: Let g be the function from the set {a,b,c}to itself such that g(a) = b, g(b) = c, and g(c) = a. Let f be the function from the set {a,b,c}to the set {1,2,3} such that f(a) = 3, f(b) = 2, and f(c) = 1. What is the composition of f and g, and what is the composition of g and f. Solution: The composition f∘g is defined by f∘g (a)= f(g(a)) = f(b) = 2. f∘g (b)= f(g(b)) = f(c) = 1. f∘g (c)= f(g(c)) = f(a) = 3. Note that g∘f is not defined, because the range of f is not a subset of the domain of g. Composition Questions Example 222: Let f and g be functions from the set of integers to the set of integers defined by f(x) = 2x + 3 and g(x) = 3x + 2. What is the composition of f and g, and also the composition of g and f ? Solution: f∘g (x)= f(g(x)) = f(3x + 2) = 2(3x +2)+ 3= 6x + 7 g∘f (x)= g(f(x)) = g(2x + 3) = 3(2x +3)+ 2= 6x + 11 Graphs of Functions ! Let f be a function from the set A to the set B. The graph of the function f is the set of ordered pairs {(a,b) | a ∈A and f(a) = b}. Graph of f(n) = 2n + 1 Graph of f(x) = x2 from Z to Z from Z to Z Some Important Functions ! The floor function, denoted is the largest integer less than or equal to x. ! The ceiling function, denoted is the smallest integer greater than or equal to x Example: Floor and Ceiling Functions Graph of (a) Floor and (b) Ceiling Functions Floor and Ceiling Functions Proving Properties of Functions Example: Prove that x is a real number, then ⌊2x⌋= ⌊x⌋ + ⌊x + 1/2⌋ Solution: Let x = n + ε, where n is an integer and 0 ≤ ε< 1. Case 1: ε < ½ ! 2x = 2n + 2ε and ⌊2x⌋ = 2n, since 0 ≤ 2ε< 1. ! ⌊x + 1/2⌋ = n, since x + ½ = n + (1/2 + ε ) and 0 ≤ ½ +ε< 1. ! Hence, ⌊2x⌋ = 2n and ⌊x⌋ + ⌊x + 1/2⌋ = n + n = 2n. Case 2: ε≥ ½ ! 2x = 2n + 2ε = (2n + 1) +(2ε − 1) and ⌊2x⌋ =2n + 1, since 0 ≤ 2 ε - 1< 1. ! ⌊x + 1/2⌋ = ⌊ n + (1/2 + ε)⌋ = ⌊ n + 1 + (ε – 1/2)⌋ = n + 1 since 0 ≤ ε – 1/2< 1. ! Hence, ⌊2x⌋ = 2n + 1 and ⌊x⌋ + ⌊x + 1/2⌋ = n + (n + 1) = 2n + 1. Factorial Function Definition: f: N → Z+ , denoted by f(n) = n! is the product of the first n positive integers when n is a nonnegative integer. f(n) = 1 ∙ 2 ∙∙∙ (n –1) ∙ n, f(0) = 0! = 1 Examples: Stirling’s Formula: f(1) = 1! = 1 f(2) = 2! = 1 ∙ 2 = 2 f(6) = 6! = 1 ∙ 2 ∙ 3∙ 4∙ 5 ∙ 6 = 720 f(20) = 2,432,902,008,176,640,000. Partial Functions (optional) Definition: A partial function f from a set A to a set B is an assignment to each element a in a subset of A, called the domain of definition of f, of a unique element b in B. ! The sets A and B are called the domain and codomain of f, respectively. ! We day that f is undefined for elements in A that are not in the domain of definition of f. ! When the domain of definition of f equals A, we say that f is a total function. Example: f: N → R where f(n) = √n is a partial function from Z to R where the domain of definition is the set of nonnegative integers. Note that f is undefined for negative integers. .
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